Riemann Sums

Approximating Area Under a Curve

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Riemann sums are a way to estimate the area under a curve by breaking an interval into many small pieces and adding the areas of simple shapes, usually rectangles. They are one of the main ideas that connect geometry and calculus. Students use them to understand accumulation, net area, and how definite integrals are built from approximations. This makes Riemann sums a foundation for later work in physics, engineering, economics, and probability.

To form a Riemann sum, you divide the interval $[a,b]$ into subintervals of width $\Delta x$ and choose a sample point in each one to determine the rectangle height. Different choices of sample points create left endpoint, right endpoint, and midpoint sums, and these can overestimate or underestimate the true area depending on the graph. As the number of subintervals increases, the rectangles become thinner and the approximation usually improves. In the limit, the Riemann sum becomes the definite integral.

Key Facts

For n equal subintervals on [a,b], delta-x = (b - a)/n.
A general Riemann sum is $\sum_{i=1}^{n} f(x_i^*)\Delta x$ , where $x_i^*$ is a sample point.
Left endpoint sum: $L_n = \sum_{i=1}^{n} f(a + (i - 1)\Delta x)\Delta x$ .
Right endpoint sum: $R_n = \sum_{i=1}^{n} f(a + i\Delta x)\Delta x$ .
Midpoint sum: $M_n = \sum_{i=1}^{n} f(a + (i - 0.5)\Delta x)\Delta x$ .
The definite integral equals the limit as $n$ approaches $\infty$ of the Riemann sum.

Vocabulary

Subinterval: One of the smaller intervals created when the interval [a,b] is divided into parts.
$\Delta x$: The width of each subinterval, often written as delta-x = (b - a)/n when all widths are equal.
Sample point: The x-value chosen inside a subinterval to determine the height of a rectangle in a Riemann sum.
Definite integral: A number that represents the accumulated signed area under a curve over an interval.
Approximation: A value that is close to the exact answer but may not be exactly equal to it.

Common Mistakes to Avoid

Using the wrong rectangle height, because students pick the midpoint when the problem asks for left or right endpoints. Always identify the sample point rule before evaluating $f(x)$ .
Forgetting to multiply by delta-x, which is wrong because adding only the heights does not give area. Each term in the sum must be height times width.
Computing delta-x incorrectly, because students sometimes divide by b - a instead of dividing the interval length by n. Use delta-x = (b - a)/n for equal subintervals.
Assuming every Riemann sum is an overestimate or underestimate, which is wrong because it depends on whether the function is increasing, decreasing, or changing shape.

Practice Questions

1 Find the left endpoint Riemann sum for $f(x) = x^2$ on $[0,2]$ with $n = 4$ equal subintervals.
2 Find the midpoint Riemann sum for $f(x) = 3x + 1$ on $[1,5]$ with $n = 4$ equal subintervals.
3 A function is increasing on [a,b]. Compare the left endpoint sum and the right endpoint sum to the true area under the curve, and explain which one is larger and why.